Cost Types

Context

• Not all costs have the same characteristics. Economists talk about fixed and variable costs and distinguish between short- and long-run horizons.

• Different costs sway (rational) business decisions in distinct ways.
  • How do fixed and variable costs differ?
  • How do short- and long-run costs differ?
  • How do these differences affect business decisions?

Course Structure Overview

Lecture Structure and Learning Objectives

Structure

• The Sunk Cost Fallacy (Case Study)
• Basic Concepts
• Examples
• A Practical Exercise
• Current Field Developments

Learning Objectives

• Explain the various types of economic costs.
• Describe the relationships between average, marginal, fixed, and total costs.
• Explain the concept of short-run costs and its relation to fixed costs.
• Describe the relationships between short- and long-run costs.
• Apply the new concepts to a practical exercise.

The Sunk Cost Fallacy

• Imagine that you and your friend have booked a wake-boarding weekend in Waldsee for \(100\) Euros each.
• Your friend approaches you after a couple of weeks and suggests you go for wake-boarding in Lautenbourg.
• You happily agree, and you book for \(50\) Euros each.
• You realize later that both bookings were for the same weekend!
• Although you like Waldsee a lot, you are sure that you will enjoy more a weekend in Lautenbourg.
• Where would you eventually choose to go?

Irrational Choices

• Many behavioral economists in the 1980s contested that humans always make rational choices, as mainstream economic theory suggests.
• Among the examples of irrational behavior was the Sunk Cost Fallacy.
• Once time, money, or effort is invested in an act or a choice, humans tend to maintain their choices.
• Economists started conducting experiments to examine if the last conjecture was true.

The Ski Trip

Assume that you have spent \($100\) on a ticket for a weekend ski trip to Michigan. Several weeks later you buy a \($50\) ticket for a weekend ski trip to Wisconsin. You think you will enjoy the Wisconsin ski trip more than the Michigan ski trip. As you are putting your just-purchased Wisconsin ski trip ticket in your wallet, you notice that the Michigan ski trip and the Wisconsin ski trip are for the same weekend! It’s too late to sell either ticket, and you cannot return either one. You must use one ticket and not the other. Which ski trip will you go on?

(Arkes & Blumer, 1985 Experiment 1)

Can this Happen in Real Life too?

• Offered discounted seasonal theater tickets to subjects and checked how often people attended. (Arkes & Blumer, 1985 Experiment 2)

• The first \(60\) people approaching the ticket window to buy a ticket were randomly split into three groups:
  • Those who paid the normal \($15\) price,
  • those who took \($2\) discount, and
  • those who took a \($7\) discount.
• Tracked the number of times each individual visited the theater.

Rational Behavior

• Mainstream economic theory suggests that rational agents should not base their decisions on sunk costs.

• Only future benefits and costs determine the optimality of decisions, as those in the past cannot change anymore.
• Acting “irrationally” at a personal level is one thing.
• As a manager, falling to suck cost fallacies can cost money and endanger your business!

Total, Variable, and Fixed Cost

A Practical Exercise

Average Total, Variable, and Fixed Cost

• Similarly, average costs can be decomposed into two parts.

• Average fixed cost is the component of the total cost that does not change with the production level (i.e., fixed cost) per unit of produced output.

• Average variable cost is the component of the total cost that changes with the production level (i.e., variable cost) per unit of produced output.

  \begin{align*} \begin{matrix}\text{Average} \\ \text{Total} \\ \text{Cost}\end{matrix} &= \begin{matrix}\text{Average} \\ \text{Variable} \\ \text{Cost}\end{matrix} + \begin{matrix}\text{Average} \\ \text{Fixed} \\ \text{Cost}\end{matrix} \\ \end{align*}

  \begin{align*} \bar{c}(q) &= \frac{c(q)}{q} \\ &= \frac{\mu(q)}{q} + \frac{\sigma(q)}{q} \\ &= \bar{\mu}(q) + \bar{\sigma}(q) \end{align*}

Marginal Cost

• The marginal cost is the rate of change of the total production cost with respect to the output level. It shows how much total cost increases when the output is marginally increased. It is given by the cost function’s derivative, i.e., \(c’\).
• Similarly, the marginal variable cost is the rate of change of the variable cost with respect to the output level. It shows how much the variable cost increases when the output is marginally increased. It is given by the derivative of the variable cost function, i.e., \(\mu’\).
• Analogously, one can define the marginal fix cost as the rate of change of the fixed cost with respect to the output level. Since fixed cost is constant, the marginal fixed cost is always zero.

Example: Calculating Costs

• Calculate the costs:

Minimizing Average Cost

Short-Run and Long-Run Costs

• The firm might not be able to adjust all production factors in short periods of time.
• E.g., some assets and properties are not always liquid.
• Some costs can then be fixed in the short-run.
• Short-run costs are greater than long run costs (why?).

Short-Run and Long-Run Cost Folding

Current Field Developments

• Well-established concepts (the cost typology) with long-standing applications.
• In merger analysis and antitrust cases, the marginal cost estimation is a central issue with many new proposed approaches in recent years.
• Recently, the sunk cost fallacy was demonstrated for non-human animals too (Sweis et al., 2018)

Comprehensive Summary

• Different cost concepts are used in economics.
• Average costs explain production costs per unit.
• Marginal costs explain the production costs of small changes in the produced output.
• Average production costs are minimized when they are equal to marginal production costs.
• Short-run costs are greater than long-run costs because some factors might not be flexible in the short-run.
• Understanding the cost types a firm deals with is important in reaching rational business decisions.

Further Reading

• Varian, 2010, Chapter 22
• CORE-Team, 2017, secs. 7.3, 7.3.1
• Arkes & Blumer, 1985

Mathematical Details

Working with the cost function has the advantage that cost can be expressed in terms of a single variable (production output). Irrespective of the number of input factors involved in the cost minimization problem, the cost function (if it exists) involves only output as a variable. This is illustrated in the following two examples.

Cost Functions Induced by Root Production Functions

Consider the cost minimization problem with a root production function

\begin{align*} \min_{x} &\left\{ w_{0} + w x \right\} \\ s.t.\quad & q \le f(x) = x^{r} \quad\quad (0 < r <1) . \end{align*}

Its solution, namely the (total) cost function, is given by

\begin{align*} c(q) = w_{0} + w q^{\frac{1}{r}}. \end{align*}

Having the cost function, one can calculate the remaining cost types as in the following table

Cost Functions Induced by Cobb-Douglas Production Functions

The cost minimization problem for a Cobb-Douglas production function is

\begin{align*} \min_{x_{1}, x_{2}} &\left\{ w_{1} x_{1} + w_{2} x_{2} \right\} \\ s.t.\quad & q \le f(x) = A \left(x_{1}^{\alpha} x_{2}^{1- \alpha}\right)^{r} \quad\quad (0 < \alpha <1, 0 < r < 1), \end{align*}

and its solution is given by

\begin{align*} c(q) = \left(\frac{q}{A}\right)^{\frac{1}{r}} w_{1}^{\alpha} w_{2}^{1 - \alpha} \left(\left(\frac{\alpha}{1 - \alpha}\right)^{1-\alpha} + \left(\frac{\alpha}{1 - \alpha}\right)^{-\alpha}\right). \end{align*}

Minimizing Average Cost

The minimum average cost is attained at the output level for which the marginal cost equals the average cost. Namely \[\min_{q} \bar{c}(q) = c’(\mathrm{arg\,min}_{q} \bar{c}(q))\] If \(c\) is differentiable, we can show this result by setting the derivative of average cost equal to zero, i.e.,

\begin{align*} \bar{c}’(q) = \frac{c’(q)q - c(q)}{q^{2}} = \frac{c’(q) - \mu(q)}{q} &\overset{!}{=} 0 \implies \\ c’(q) &\overset{!}{=} \mu(q) \end{align*}

Analogously, we can show that the minimum average variable cost is attained at the output level for which the marginal cost equals the average variable cost. \[\min_{q} \bar{\mu}(q) = c’(\mathrm{arg\,min}_{q} \bar{\mu}(q)).\]

The Short- and Long-Run Relationship

Consider again the cost minimization problem

\begin{align*} \min_{x_{1}, x_{2}} &\left\{ w_{1} x_{1} + w_{2} x_{2} \right\} \\ s.t.\quad & q \le f(x) = A \left(x_{1}^{\alpha} x_{2}^{1- \alpha}\right)^{r} \quad\quad (0 < \alpha <1, 0 < r < 1) , \end{align*}

with resulting cost function

\begin{align*} c(q) = \left(\frac{q}{A}\right)^{\frac{1}{r}} w_{1}^{\alpha} w_{2}^{1 - \alpha} \left(\left(\frac{\alpha}{1 - \alpha}\right)^{1-\alpha} + \left(\frac{\alpha}{1 - \alpha}\right)^{-\alpha}\right) . \end{align*}

What happens if one production function, say \(x_{2}\), is fixed? It is common for many production settings that some factors cannot be flexibly adjusted in short periods. This results in a fixed cost component. Fixing the level of \(x_{2} = \hat{x}_{2}\), the minimization problem becomes

\begin{align*} \min_{x_{1}} &\left\{ w_{1} x_{1} + \underbrace{w_{2} \hat{x}_{2}}_{w_{0}} \right\} , \\ s.t.\quad & q = f(x_{1}) = \underbrace{\left(A \hat{x}_{2}^{(1- \alpha)r}\right)}_{B} x_{1}^{\overbrace{\alpha r}^{\beta}} \quad\quad (0 < \alpha <1, 0 < r < 1) . \end{align*}

Its solution is given by

\begin{align*} c(q; \hat{x}_{2}) = \left(\frac{q}{A}\right)^{\frac{1}{\alpha r}} w_{1} \hat{x}_{2}^{\frac{\alpha-1}{\alpha}} + w_{2} \hat{x}_{2} . \end{align*}

Using the \(\beta\), \(B\) notation, we observe that the production function essentially becomes a root function (\(f(x_{1}) = B x_{1}^{\beta}\)), and the resulting cost function is very similar to the cost function obtained by single variable, root production functions.

Cost folding

When all factors are allowed to vary, the cost is lower or equal than when at least one factor is fixed. Since \(x_{2}\) is chosen to minimize costs if it is allowed to vary, it must hold

\begin{align*} c(q) \le c(q; \hat{x}_{2}). \end{align*}

Equality is attained when \(x_{2}\) is fixed to the optimal conditional demanded quantity given the output level \(q\), i.e.

\begin{align*} c(q) = c(q; x_{2}(q)) . \end{align*}

Exercises

Group A

1. Consider the cost function \(c(q) = 16 + 4q^{2}\). Calculate and plot

   1. the average cost function,
   2. the marginal cost function,
   3. the average variable cost function.
   4. the average fixed cost function.
   5. What is the level of output that yields the minimum average production cost?

   Cost type	Expression
   Total cost	\(c(q) = 16 + 4 q^{2}\)
   Average cost	\(\bar{c}(q) = \frac{16}{q} + 4 q\)
   Average variable cost	\(\bar{\mu}(q) = 4 q\)
   Average fixed cost	\(\bar{\sigma}(q) =\frac{16}{q}\)
   Marginal cost	\(c’(q) = 8 q\)

   The minimum average cost is obtained by the first order condition \[ -\frac{16}{q^{2}} + 4 = 0, \] which implies that \(q = 2\).

2. Consider the cost function obtained by a Cobb-Douglas production function

   \begin{align*} c(q; w_{1}, w_{2}) &= \left(\frac{\alpha}{w_{1}}\right)^{-\alpha} \left(\frac{1 - \alpha}{w_{2}}\right)^{-(1 - \alpha)} \left(\frac{q}{A}\right)^{\frac{1}{r}}, \end{align*}

   and let \(w_{1}=w_{2}=\alpha = 1 / 2\), \(A=1\), and \(r>0\). Show that the average cost function is

   1. increasing for decreasing returns to scale (\(r<1\)),
   2. decreasing for increasing returns to scale (\(r>1\)), and
   3. constant for constant returns to scale (\(r=1\)).

   The average cost function for the given parameters values is given by

   \begin{align*} \bar{c}(q) &= q^{\frac{1 - r}{r}}, \end{align*}

   with derivative

   \begin{align*} \bar{c}’(q) &= \frac{1 - r}{r}q^{\frac{1 - 2r}{r}}. \end{align*}

   If \(r<1\), then \(\bar{c}’ >0\) and \(\bar{c}\) is increasing. If \(r>1\), then \(\bar{c}’ <0\) and \(\bar{c}\) is decreasing. Finally, if \(r=1\), then \(\bar{c}’ =0\) and \(\bar{c}\) is constant.

References

References

Topic's Concepts

• marginal fix cost
• marginal variable cost
• marginal cost
• average variable cost
• average fixed cost
• variable cost